Find the optical power and the focal lengths
(a) of a thin glass lens in liquid with refractive index `n_(0) = 1.7` if its optical power in air is `Phi_(0) = 5.0 D`,
(b) of a thin symmetrical biconvex glass lens, with air on one side and water on the other side, if the optical power of that lens in air is `Phi_(0) = + 10D`.

Explanation

Optical power of a thin lens of `R.I. n` in a medium with `R.I. n_(0)` is given by :
`Phi = (n - n_(0)) ((1)/(R_(1)) - (1)/(R_(2)))` (A)
From Eqn. (A) when the lens is placed in air :
`Phi_(0) = (n - 1) ((1)/(R_(1)) - (1)/(R_(2)))` (1)
Similarly from Eqn.(A), when lens is placed in liquid :
`Phi = (n - n_(0)) ((1)/(R_(1)) - (1)/(R_(2)))` (2)
Thus from Eqn (1) and (2)
`Phi = (n - n_(0))/(n - 1) Phi_(0) = 2D`
The second focal length, is given by
`f' = (n')/(Phi)`, where `n`' is the `R.I.` of the medium in which it is placed.
`f' = (n_(0))/(Phi) = 85 cm`
(b) Optical power of a thin lens of `R.I. n` placed in a medium of `R.I. n_(0)` is given by :
`Phi = (n - 1) ((1)/(R_(1)) - (1)/(R_(2)))` (A)
For a biconvex lens placed in air medium from Eqn(A)
`Phi_(0) = (n - 1) ((1)/(R) - (1)/(-R)) = (2(n - 1))/(R )` (1)
where `R` is the radius of each curve surface of the lens
Optical power of a spherical refractive surface is given by :
`Phi = (n' - n)/(R )` (B)
For the rear surface of the lens which divides air and glass medium
`Phi_(0) = (n - 1)/(R )` (Here `n` is the `R.I.`(2) of glass)
Similarly fro the front surface which divides water and glass medium
`Phi_(l) = (n - n)/(-R ) = (n - n_(0))/(R )` (3)
Hence the optical power of the given optical system
`Phi = Phi_(e) + Phi_(l) = (n - 1)/(R ) + (n - n_(0))/(R ) = (2n - n_(0) - 1)/(2(n - 1)) Phi_(0)`
Focal length in air, `f = (1)/(Phi) = 15 cm`
and focal length in water `= (n_(0))/(Phi) = 20 cm` for `n_(0) = (4)/(3)`.